Two-level methods based on the Arrow–Hurwicz iteration for the steady incompressible magnetohydrodynamic system

We consider the two-level methods based on Arrow–Hurwicz (A-H) iteration for solving the stationary incompressible magnetohydrodynamics problem. The methods carry out some A-H iterations on the coarse mesh, improve the obtained solutions by using one-step scheme on the fine mesh. Such methods enjoy some advantages including that the number of A-H iterations can be flexibly and controllably selected, and they are valid for high hydrodynamic Reynolds number. The convergence analyses are established and various numerical experiments are performed to validate and illustrate the stability, accuracy, and efficiency of the new methods.